Biology Reference
In-Depth Information
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4 partial warp scores, but this no longer matches the degrees of freedom in the data.
Therefore, other approaches to dimensionality reduction must be used, such as Principal
Components Analysis.
The spline allows a decomposition of a bending energy matrix describing the differ-
ences into partial warps, which provides such a set of orthogonal vectors; the partial warps
supply a basis for the tangent space of all possible shape differences relative to the refer-
ence. Unlike the coordinates obtained by the Procrustes-based superimposition methods,
the thin-plate spline coefficients (called partial warp scores) can be used in conventional sta-
tistical tests without adjusting the degrees of freedom, so long as the data contains no
semilandmarks, as noted above. Also unlike the coordinates produced by the two-point
registration, which also have the appropriate number for statistical tests, the partial warp
scores enable using the correct tangent space measure of distance
the partial Procrustes
distance. Changing from one orthonormal basis to another does not alter the distances in
this linear tangent space to shape space. Using partial warp scores you will get the
same distances between specimens as you get using the coordinates obtained by the
Procrustes (GPA) superimposition, in a linear tangent space approximation. To get
the same statistical results, you do need to adjust correctly the degrees of freedom present in
the landmark data.
Additionally, the thin-plate spline provides an approach for superimposing (sliding)
semilandmarks. One approach to semilandmark alignment is to minimize the distance
between corresponding semilandmarks, alternatively, the approach based on the thin-plate
spline slides the semilandmarks to produce the smoothest (least localized) deformation of
one curve into another. The thin-plate spline method thus provides a smooth deformation
criterion for semilandmarks.
In summary, the thin-plate spline provides a visually interpretable description of a
deformation, with the same number of variables as there are statistical degrees of freedom
so long as the data consist solely of landmarks. Even if we were not concerned with the
advantages of the spline for graphical analysis, nor wished to use it for sliding semiland-
marks, we might still want to use the partial warps for purposes of statistical inference.
Many of the popular programs for statistical shape analysis use partial warp scores in
their internal calculations, although this may change in the future as semilandmarks are
increasingly used. Even if we were not concerned with the advantages of the partial warps
for statistical analysis, we might still wish to use the thin-plate spline for its graphical
capabilities or to slide semilandmarks. You can use the spline to depict your results, and
you can use partial warps in your statistical analyses without worrying that the mathemat-
ical details (and complexities) will have any impact on your results, although you will
have to be aware of the difficulties posed by the dimensionality of semilandmarks. The
spline is a convenient tool for visual display and for obtaining variables with the correct
degrees of freedom
it is nothing more (or less) than that.
In this chapter, we begin with a basic overview of the mathematical idea of a deforma-
tion. We then discuss the mathematical metaphor underlying one particular model of a
deformation, the thin-plate spline, and how we can decompose it to yield variables. In
general, we present a largely intuitive overview before delving more deeply into the math-
ematics. At the end of this chapter, we summarize a method for sliding semilandmarks
based on the thin-plate spline.
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